Sequential fully implicit well modeling of transmissibility for reservoir simulation

ABSTRACT

A subsurface hydrocarbon reservoir with wells is simulated by sequential solution of reservoir and well equations to simulate fluid flow inside the reservoir and well production rates. Sequential solution of reservoir and well equations treats wells as specified bottom hole pressure wells. This avoids solving large matrices resulting from the simultaneous solution of the reservoir and well equations which can be computationally very expensive for large number of unknowns and require special sparse matrix solvers. Such sequential solution involves regular reservoir system solvers complemented by a small matrix for the numerical solution of the well bottom hole pressures. The solution is performed on a matrix of the transmissibilities of the adjacent reservoir cells to the well cells at the perforated well intervals; and a vector of unknown reservoir potentials for the adjacent reservoir cells.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of, and claims priority to, Applicant's co-pending, commonly owned U.S. patent application Ser. No. 14/040,930, filed Sep. 30, 2013, now U.S. Pat. No. 9,494,709, issued Nov. 15, 2016; and to U.S. patent application Ser. No. 13/023,728, filed Feb. 9, 2011, now U.S. Pat. No. 9,164,191, issued Oct. 20, 2015.

The present application is also related to commonly owned U.S. patent application Ser. No. 15/061,472, filed of even date herewith, entitled “Sequential Fully Implicit Well Modeling of Transmissibility For Reservoir Simulation,” and having the same inventor as the present application.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to computerized simulation of hydrocarbon reservoirs in the earth, and in particular to simulation of flow profiles along wells in a reservoir.

2. Description of the Related Art

Well models have played an important role in numerical reservoir simulation. Well models have been used to calculate oil, water and gas production rates from wells in an oil and gas reservoirs. If the well production rate is known, they are used to calculate the flow profile along the perforated interval of the well. With the increasing capabilities for measuring flow rates along the perforated intervals of a well, a proper numerical well model is necessary to compute the correct flow profile to match the measurements in a reservoir simulator.

It is well known that simple well models such as explicit or semi implicit models could be adequate if all reservoir layers communicated vertically for any vertical wells in a reservoir simulator. For these models, well production rate is allocated to the perforations in proportion to the layer productivity indices (or total mobility). Therefore, the calculations are simple and computationally inexpensive. The structure of the resulting coefficient matrix for the reservoir unknowns remains unchanged. Specifically, the coefficient matrix maintains a regular sparse structure. Therefore, any such sparse matrix solver could be used to solve the linear system for the grid block pressures and saturations for every time step for the entire reservoir simulation model.

However, for highly heterogeneous reservoirs with some vertically non-communicating layers, the above-mentioned well models cannot produce the correct physical solution. Instead, they produce incorrect flow profiles and in some occasions cause simulator convergence problems.

With the increasing sophistication of reservoir models, the number of vertical layers has come to be in the order of hundreds to represent reservoir heterogeneity. Fully implicit, fully coupled well models with simultaneous solution of reservoir and well equations have been necessary to correctly simulate the flow profiles along the well and also necessary for the numerical stability of the reservoir simulation. In order to solve the fully coupled system, generally well equations are eliminated first. This creates an unstructured coefficient matrix for the reservoir unknowns to be solved. Solutions of this type of matrices require specialized solvers with specialized preconditioners. For wells with many completions and many wells in a simulation model, this method has become computationally expensive in terms of processor time.

SUMMARY OF THE INVENTION

Briefly, the present provides a new and improved computer implemented method of solving well equations together with reservoir equations in a reservoir simulation model having formation layers having vertical fluid flow and flow barrier layers with no vertical fluid flow. The computer implemented method forms a reduced well model system for the wells in a reservoir simulation model by assembling as a single vertical flow layer the flow layers having vertical flow between the vertical flow barriers in the reservoir model in the vicinity of the wells only. For a production rate specified well, the method then solves the reduced well model system by matrix computation (using a direct sparse matrix solver) for the bottom hole pressure of the well and reservoir unknowns for the grid blocks where well is going through (reduced well model system). This method is repeated for all the wells in a reservoir simulator. Next, the method then solves the full reservoir simulation model by treating each well as having the determined bottom hole pressure, and based The solution is performed on a matrix of the transmissibilities of the adjacent reservoir cells to the well cells at the perforated well intervals; and a vector of unknown reservoir potentials for the adjacent reservoir cells to determine completion rates for the layers of the well for the full reservoir model, and determine total rate for the well from the determined completion rates for the layers of the well. The method then forms a record of the determined completion rates for the layers and the determined total rate for the well. The simulator then calculates the pressure distribution, if multi-phase fluid saturations distribution in the reservoir by using the calculated perforation rates and the reservoir data assigned to the simulator grid blocks. The simulator is thus able to calculate pressure and saturation distribution within the reservoir with given production and injection rates at the wells at a given time.

The present invention provides a new and improved data processing system for forming a well model for reservoir simulation of well in a subsurface reservoir from a reservoir model having formation layers having vertical fluid flow and flow barrier layers with no vertical fluid flow. The data processing system includes a processor which performs the steps of solving by matrix computation the reduced well model system model for the bottom hole pressure of the well and reservoir unknowns around the well and solving the full reservoir model by treating the well as having the determined bottom hole pressure, and based on a matrix of the transmissibility of the adjacent reservoir cells to the well cells at the perforated well intervals; and a vector of unknown reservoir potentials for the adjacent reservoir cells to completion rates for the layers of the well for the full reservoir model and the pressure and the fluid saturation distribution in the reservoir at any grid block at any time. Calculated perforation rates are calculated from the specified well production or injection rates using well models in order to calculate the reservoir pressure and saturation distribution at any time during the life of the reservoir or in the future. The processor also determines total rate for the well from the determined completion rates for the layers of the well. The data processing system also includes a memory forming a record the determined completion rates for the layers and the determined total rate for the well.

The present invention further provides a new and improved data storage device having stored in a computer readable medium computer operable instructions for causing a data processor in forming a well model for reservoir simulation of well in a subsurface reservoir from a reservoir model having formation layers having vertical fluid flow and flow barrier layers with no vertical fluid flow to perform steps of solving by matrix computation the reduced well model system model for the bottom hole pressure of the well and reservoir unknowns and solving the full reservoir model by treating the well as having the determined bottom hole pressure, based on a steady state volume balance relationship of the layer completion rates, formation pressures and transmissibilities to determine completion rates for the layers of the well for the full reservoir model. The instructions stored in the data storage device also include instructions causing the data processor to determine the total rate for the well from the determined completion rates for the layers of the well, and form a record the determined completion rates for the layers and the determined total rate for the well.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B are schematic diagrams for a well model in a reservoir simulator of multiple subsurface formation layers above and below a flow barrier in a reservoir being formed into single layers according to the present invention.

FIGS. 2A and 2B are schematic diagrams for a well model in a reservoir simulator of multiple subsurface formation layers above and below several vertically spaced flow barriers in a reservoir being formed into single layers according to the present invention.

FIG. 3 is a schematic diagram of a well model for simulation based on an explicit model methodology illustrated in radial geometry.

FIG. 4 is a schematic diagram of a well model for simulation based on a fully implicit, fully coupled model methodology.

FIG. 5 is a schematic diagram of a well model for reservoir simulation with a comparison of flow profiles obtained from the models of FIGS. 3 and 4.

FIG. 6 is a schematic diagram of a finite difference grid system for the well models of FIGS. 3 and 4.

FIGS. 7A and 7B are schematic diagrams illustrating the well model reservoir layers for an unmodified conventional well layer model and a well layer model according to the present invention, respectively.

FIGS. 8A and 8B are schematic diagrams of flow profiles illustrating comparisons between the models of FIGS. 7A and 7B, respectively.

FIG. 9 is a schematic diagram of a well layer model having two fracture layers in radial coordinates.

FIG. 10 is a functional block diagram or flow chart of data processing steps for a method and system for a fully implicit fully coupled well model with simultaneous numerical solution for reservoir simulation.

FIG. 11 is a functional block diagram or flow chart of data processing steps for a method and system for a sequential fully implicit well model for reservoir simulation according to the present invention.

FIG. 12 is a schematic diagram of a linear system of equations with a tridiagonal coefficient matrix for an explicit well model for a one dimensional reservoir simulator with a vertical well

FIG. 13 is a schematic diagram of a linear system of equations with a tridiagonal coefficient matrix for a fully implicit well model for a one dimensional reservoir simulator with a vertical well

FIG. 14 is a schematic diagram of a linear system of equations with a tridiagonal coefficient matrix for a constant bottom hole pressure well model for a one dimensional reservoir simulator with a vertical well for processing according to the present invention.

FIG. 15 is a functional block diagram or flow chart of steps illustrating the analytical methodology for reservoir simulation according to the present invention.

FIG. 16 is a schematic diagram of a computer network for a sequential fully implicit well model for reservoir simulation according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

By way of introduction, the present invention provides a sequential fully implicit well model for reservoir simulation. Reservoir simulation is a mathematical modeling science for reservoir engineering. The fluid flow inside the oil or gas reservoir (porous media) is described by a set of partial differential equations. These equations describe the pressure (energy) distribution, oil, water and gas velocity distribution, fractional volumes (saturations) of oil, water, gas at any point in reservoir at any time during the life of the reservoir which produces oil, gas and water. Fluid flow inside the reservoir is described by tracing the movement of the component of the mixture. Amounts of components such as methane, ethane, CO₂, nitrogen, H₂S and water are expressed either in mass unit or moles.

Since these equations and associated thermodynamic and physical laws describing the fluid flow are complicated, they can only be solved on digital computers to obtain pressure distribution, velocity distribution and fluid saturation or the amount of component mass or mole distribution within the reservoir at any time at any point. This only can be done by solving these equations numerically, not analytically. Numerical solution requires that the reservoir be subdivided into computational elements (cells or grid blocks) in the area and vertical direction (x, y, z—three dimensional space) and time is sub-divided into intervals of days or months. For each element, the unknowns (pressure, velocity, volume fractions, etc.) are determined by solving the complicated mathematical equations.

In fact, a reservoir simulator model can be considered the collection of rectangular prisms (like bricks in the walls of a building). The changes in the pressure and velocity fields take place due to oil, water and gas production at the wells distributed within the reservoir. Simulation is carried out over time (t). Generally, the production or injection rate of each well is known during the production history of a reservoir. However, since the wells go through several reservoir layers (elements), the contribution of each reservoir element (well perforation) to the production is calculated by different methods. This invention deals with the calculation of how much each well perforation contributes to the total well production. Since these calculations can be expensive and very important boundary conditions for the simulator, the proposed method suggests a practical method to calculate correctly the flow profiles along a well trajectory. As will be described, it can be shown that some other methods used will result in incorrect flow profiles which cause problems in obtaining the correct numerical solution and can be very expensive computationally.

Nomenclature

-   -   Δx, Δy, Δz=Grid dimension in x, y and z directions     -   k_(x), k_(y), k_(z)=permeability in x, y and z directions     -   p=pressure     -   ρ=Fluid (oil) density     -   g=gravitational constant     -   z=vertical depth from a datum depth     -   r_(o)=Peaceman's radius=0.2 Δx     -   r_(w)=wellbore radius     -   T_(x), T_(y), T_(z)=Rock Transmissibility in the x, y and z         directions

The equations describing a general reservoir simulation model and indicating the well terms which are of interest in connection with the present invention are set forth below in Equation (1):

$\begin{matrix} {{{{\Delta_{x}{\sum\limits_{j = 1}^{n_{p}}{T_{x}\rho_{ij}\lambda_{j}\mspace{11mu}{\Delta\Phi}_{j}}}} + {\Delta_{y}{\sum\limits_{j = 1}^{n_{p}}{T_{y}\rho_{ij}\lambda_{j}\mspace{11mu}{\Delta\Phi}_{j}}}} + {\Delta_{z}{\sum\limits_{j = 1}^{n_{p}}{T_{z}\rho_{ij}\lambda_{j}\mspace{11mu}{\Delta\Phi}_{j}}}} + {\sum\limits_{j = 1}^{n_{p}}q_{ij}}} = \frac{\Delta_{t}n_{i}}{\Delta\; t}}\mspace{20mu}{{i = 1},{\ldots\mspace{14mu} n_{c}\mspace{14mu}\ldots}}} & (1) \end{matrix}$ where Tx, Ty, Tz are the rock transmissibilities in the x, y and z directions as defined in Equation below), j stands for the number of fluid phases, n_(p) is the total number of fluid phases, usually 3 (oil, water and gas), Σ is the summation term, ρ_(ij) is the density of component i in the fluid phase j, λ_(j) is the mobility of the phase j (Equation 6), Φ_(j) is the fluid potential (datum corrected pressure) of fluid phase j, similarly Δ_(x) is the difference operator in x direction, and Δ_(z) is the difference operator in z direction of the reservoir, q_(ij) is the volumetric well term (source or i sink) for the component i in phase j for grid block (cell) located at (x, y, z), Δ_(t) is the difference operator in time domain, n_(i) is the total number of moles for component i, and n_(c) is the total number of components in the fluid system (methane, ethane, propane, CO₂, etc.).

Equation (1) is a set of coupled nonlinear partial differential equations describing the fluid flow in the reservoir. In the above set of equations i represents the ith component of the fluids. nc is the total number of components of the hydrocarbons and water flowing in the reservoir. Here a component means such as methane, ethane, propane, CO2, H2S, water, etc. The number of components depends on the hydrocarbon water system available for the reservoir of interest. Typically, the number of components can change from 3 to 10. Equation (1) combines the continuity equations and momentum equations. In Equation (1) q_(i,j) is the well perforation rate at location x, y, x for component i and x, y, z is the center of grid block k (cell) number. Again, the calculation of this term from the specified production rates at the well head is the subject of the present invention.

In addition to the differential equations in Equation (1), pore volume constraint at any point (element) in the reservoir must be satisfied:

$\begin{matrix} {{V_{p}\left( {p\left( {x,y,z} \right)} \right)} = {\sum\limits_{j = 1}^{n_{p}}\frac{N_{j}}{\rho_{j}}}} & (2) \end{matrix}$ where V_(p) is the grid block pore volume, p (x, y, z) is the fluid pressure at point x, y, z, N_(j) is the total number of moles in fluid phase j, ρ_(j) is the density of fluid phase j.

There are n_(c)+1 equations in Equations (1) and (2), and n_(c)+1 unknowns. These equations are solved simultaneously with thermodynamics phase constraints for each component i by Equation (3): f _(i) ^(V)(n _(iV) ;n ₁ ,n ₂ , . . . n _(c) ,p,T)=f _(i) ^(L)(n _(iL) ;n ₁ ,n ₂ , . . . n _(c) ,p,T)  (3) where f_(i) is the component fugacity, superscript V stands for the vapor phase, L stands for the liquid phase, n₁ is the total number of moles of component i, p is the pressure and T represents the temperature.

In the fluid system in a reservoir typically there are three fluid phases: oil phase, gas phase and water phase. Each fluid phase may contain different amounts of components described above based on the reservoir pressure and temperature. The fluid phases are described by the symbol j. The symbol j has the maximum value 3 (oil, water and gas phases). The symbol n_(p) is the number of phases (sometimes it could be 1 (oil); 2 (oil and gas or oil and water); or 3 (oil, water and gas)). The number of phases n_(p) varies based on reservoir pressure (p) and temperature (T). The symbol n_(i) is the number of moles of component i in the fluid system. The symbol n_(c) is the maximum number of components in the fluid system. The number of phases and fraction of each component in each phase n_(i,j) as well as the phase density ρ_(i) and ρ_(i,j) are determined from Equation (3). In Equation (3), V stands for the vapor (gas) phase and L stands for liquid phase (oil or water).

Total number of moles in a fluid phase j is defined by: N _(j)=Σ_(i=1) ^(n) ^(c) n _(i,j)  (4)

Total component moles are defined by Equation (5). N _(i)=Σ_(j=1) ^(n) ^(p) n _(i,j)  (5)

Phase mobility in Equation (1), the relation between the phases, definition of fluid potential and differentiation symbols are defined in Equations (6) through (9). λ_(j) =k _(r,j)/μ_(j)  (6)

In Equation (6), the numerator defines the phase relative permeability and the denominator is the phase viscosity.

The capillary pressure between the phases are defined by Equation (7) with respect to the phase pressures: P _(c)(s _(j) ,s _(j′))=P _(j) −P _({tilde over (j)}′)  (7)

The fluid potential for phase j is defined by: Φ_(j) =P _(j) −gρ _(j) z  (8)

Discrete differentiation operators in the x, y, and z directions are defined by: Δ_(x) U _(x+Δx) −U _(x), Δ_(y) =U _(y+Δy) −U _(y), Δ_(z+Δz) =U _(z+Δz) −U _(z) Δ_(t) =U _(t+Δt) −U _(t),  (9) where U is any independent variable.

Equations (1), together with the constraints and definitions in Equations (3) through (9), are written for every grid block (cell) in a reservoir simulator using control volume finite difference method (some of the grid blocks may include wells). Resulting equations are solved simultaneously. This is done to find the distribution of n_(i) (x, y, x, and t), p (x, y, z, and t) for the given well production rates q_(T) for each fluid phase well from which the component rates in Equation (1) are calculated according to the present invention using the new well model formulation. In order to solve Equations (1) and (2), reservoir boundaries in (x, y, z) space, rock property distribution K (x, y, z), rock porosity distribution and fluid properties and saturation dependent data is entered into simulation.

According to the present invention and as will be described below, a reduced well model system is formed which yields the same determination of a calculated bottom hole pressure as complex, computationally time consuming prior fully coupled well models.

According to the present invention, it has been determined that for the grid blocks where the well trajectory is going through where a number of formation layers communicate vertically, the communicating layers can be combined for processing into a single layer, as indicated schematically in FIGS. 1 and 2 for the well model. This is done by identifying the vertical flow barriers in the reservoir for the well cells, and combining the layers above and below the various flow barriers of the well cells. Therefore, the full well model system is reduced to a smaller dimensional well model system with many fewer layers for incorporation into a well model for processing.

As shown in FIG. 1, a well model L represents in simplified schematic form a complex subsurface reservoir grid blocks (cells) where the well is going through which is composed of seven individual formation layers 10, each of which is in flow communication vertically with adjacent layers 10. The model L includes another group of ten formation layers around the well 12, each of which is in flow communication vertically with adjacent layers 12. The groups of formation layers 10 and 12 in flow communication with other similar adjacent layers in model L are separated as indicated in FIG. 1 by a fluid impermeable barrier layer 14 which is a barrier to vertical fluid flow.

According to the present invention, the well model L is reduced for processing purposes to a reduced or simplified well model R (FIG. 1A) by lumping together or combining, for the purposes of determining potential Φ and completion rates, the layers 10 of the well model L above the flow barrier 14 into a composite layer 10 a in the reduced model R. Similarly, the layers 12 of the well model L below the flow barrier 14 are combined into a composite layer 12 a in the reduced well model R.

Similarly, as indicated by FIG. 2, a reservoir model L−1 is composed of five upper individual formation layers 20, each of which is in flow communication vertically with adjacent layers 20. The model L−1 includes another group of seven formation layers 22, each of which is in flow communication vertically with adjacent layers 22. The groups of formation layers 20 and 22 in flow communication with other similar adjacent layers in model L−1 are separated as indicated in FIG. 2 by a fluid impermeable barrier layer 24 which is a barrier to vertical fluid flow. Another group of nine formation layers 25 in flow communication with each other are separated from the layers 22 below a fluid barrier layer 26 which is a vertical fluid flow barrier as indicated in the model L−1. A final lower group of ten formation layers 27 in flow communication with each other are located below a fluid flow barrier layer 28 in the model L−1.

According to the present invention, the well model L−1 is reduced for processing purposes to a reduced or simplified model R−1 (FIG. 2A) by lumping together or combining, for the purposes of determining well layer potential Φ and completion rates, the layers 20 of the model L−1 above the flow barrier 24 into a composite layer 20 a in the reduced model R. Similarly, other layers 22, 25, and 27 of the model L−1 below flow barriers 24, 26 and 28 are combined into composite layers 22 a, 25 a, and 27 a in the reduced model R−1.

The reduced well model systems or well models according to the present invention are solved for the reservoir unknowns and the well bottom hole pressure. Next, the wells in the full reservoir simulation model system are treated as specified bottom hole well pressure and solved implicitly for the reservoir unknowns. The diagonal elements of the coefficient matrix and the right hand side vector for the reservoir model are the only components which are modified in processing according to the present invention, and these are only slightly modified. A regular sparse solver technique or methodology is then used to solve for the reservoir unknowns. The perforation rates are computed by using the reservoir unknowns (pressures and saturations). These rates are then summed up to calculate the total well rate. The error between the determined total well rate according to the present invention and the input well rate will diminish with the simulator's non-linear Newton iterations for every time step.

The flow rates calculated according to the present invention also converge with the flow rates calculated by the fully coupled simultaneous solution for the entire reservoir simulation model including many wells. Because the present invention requires solving a small well system model, the computational cost is less. It has been found that the methodology of the present invention converges if the reduced well system is constructed properly, by using upscaling properly when combining communicating layers.

It is well known that simple vertical well models such as explicit or semi-implicit models have been generally adequate if all reservoir layers communicate vertically. As shown in FIG. 3, an explicit well model E is composed of a number Nz of reservoir layers 30 in vertical flow communication, each layer having a permeability k_(x,i) (here i is the layer number—not component) and a thickness Δz_(i) and a perforation layer rate q_(i) defined as indicated in FIG. 3. The total production rate q_(T) for the explicit model E is then the sum of the individual production rates q_(i) for the Nz layers of the explicit model as indicated in Equation (3) in the same Figure.

For explicit models, the well production rate is allocated to the perforations in proportion to the layer productivity indices (or total mobility). Therefore, the calculations are simple. The resulting coefficient matrix for the unknowns remains unchanged, i.e., maintains a regular sparse structure, as shown in matrix format in FIG. 12. Therefore, any sparse matrix solver can be used to solve the linear system for the grid block pressures and saturations for every time step.

Well Models

The methodology of several vertical well models of a reservoir simulator is presented based also for simplicity on a simple fluid system in the form of flow of a slightly compressible single phase oil flow inside the reservoir. However, it should be understood that the present invention general in applicability to reservoirs, and can be used for any number of wells and fluid phases in a regular reservoir simulation model.

FIG. 6 illustrates the finite difference grid G used in this description for a vertical well model. As seen, a well is located at the center of the central cell in vertical directions. The models set forth below also contemplate that the well is completed in the vertical Nz directions, and the potentials in the adjacent cells: Φ_(BW),Φ_(BE),Φ_(BN),Φ_(BS) are constants and known from the simulation run (previous time step or iteration value). Here subscript B refers to “Boundary”, W indicates west neighbor, E stands for East neighbor, and N and S stand for north and south respectively. Again Φ describes the fluid potential (datum corrected pressure).

As can be seen in FIG. 6, the well is penetrating a number of reservoir layers in the vertical direction represented by the index I, for i=1, 2, 3 . . . Nz, with Nz is being the total number of vertical layers in the reservoir model. For each layer i, there are four neighboring cells at the same areal plane (x, y). These neighboring cells are in the East-West (x direction) and North-South (y direction).

The potential Φ of the East, West, North and South neighbors are known from the simulator time step calculations, and those potentials are set in that they are assumed not to change over the simulator time step. The vertical potential variations at the well location are then considered, but neighbor potentials which can vary in the vertical direction but are assumed to be known.

The steady state volume balance equation for the cell (i) in FIG. 6 is as follows: T _(wi)(Φ_(Bw)−Φ_(i))+T _(Ei)(Φ_(BE)−Φ_(i))+T _(Ni)(Φ_(BN)−Φ_(i))+T _(Si)(Φ_(BS)−Φ_(i))+T _(Up,i)(Φ_(i−1)−Φ_(i))+T _(Down,i)(Φ_(i−1)−Φ_(i))−q _(i)=0.  (10)

In Equation (10), T represents the transmissibility between the cells. The subscripts W, E, N, and S denote west, east, south and north directions, and (i) represent the cell index.

The transmissibilities between cells for three directions are defined by Equation (11) below:

$\begin{matrix} {{T_{wi} = {k_{x,{i - {1/2}}}\frac{\Delta\; y_{i}\Delta\; z_{i}}{0.5\left( {{\Delta x}_{i - 1} + {\Delta\; x_{i}}} \right)}}}{T_{E,i} = {k_{x,{i + \frac{1}{2}}}\frac{\Delta\; y_{i}\Delta\; z_{i}}{0.5\left( {{\Delta\; x_{i + 1}} + {\Delta\; x_{i}}} \right)}}}{T_{Ni} = {k_{y,{i - {1/2}}}\frac{\Delta\; x_{i}\Delta\; z_{i}}{0.5\left( {{\Delta\; y_{i - 1}} + {\Delta\; y_{i}}} \right)}}}{T_{S,i} = {k_{y,{i + \frac{1}{2}}}\frac{\Delta\; x_{i}\Delta\; z_{i}}{0.5\left( {{\Delta\; y_{i + 1}} + {\Delta\; y_{i}}} \right)}}}{T_{{Up},i} = {k_{z,{i - {1/2}}}\frac{\Delta\; x_{i}\Delta\; y_{i}}{0.5\left( {{\Delta\; z_{i - 1}} + {\Delta\; z_{i}}} \right)}}}{T_{{Down},i} = {k_{z,{i + {1/2}}}\frac{\Delta\; x_{i}\Delta\; y_{i}}{0.5\left( {{\Delta\; z_{i + 1}} + {\Delta\; z_{i}}} \right)}}}} & (11) \end{matrix}$

In the above Equations (11), Δx_(i) is the grid block size (cell size) in the x direction for the grid block(cell) number i. Similarly, Δy_(i) is the grid block size (cell size) in the y direction for the grid block(cell) number I, and Δz_(i) is the grid block size (cell size) or grid layer thickness in the z direction for the grid block(cell) number i. K_(z,i−1/2) is the vertical permeability at the interface of the cells i and i−1. Similarly, K_(z,i+1/2) is the vertical permeability at the interface of the cells i and i+1. As seen in FIG. 6, the cell j is at the center, i+1/2 interface between the cell i and i+1. For convenience the subscript j is dropped while expressing East-West flow. Similarly (i, j−1) is the north neighbor of the central cell (i, j). Therefore, the notation (i, j−½) is the interface between the central cell and north neighbor in the y direction.

The same notation is carried out for the south neighbor: (i, j+½) denotes the interface between the central cell (i, j) and south neighbor (i, j+1), in the y direction. For convenience in the above equations, subscript i is dropped while expressing y (or j) direction. As is evident, transmissibilities in Equation (11) are defined in a similar manner.

In FIG. 14, the diagonal terms {tilde over (T)}_(c,i) are defined by Equation (17a) below, and the transmissibilities between cells for three directions are as defined in Equations 11, as set forth above.

The term {tilde over (b)}_(i) in the FIG. 14 (right hand side) are expressed by Equation (17b) in the text. It is extracted here as follows: {tilde over (b)} _(i)=−(PI_(i)Φ_(w) +T _(wi)Φ_(Bw) +T _(Ei)Φ_(BE) +T _(Ni)Φ_(BN) +T _(Si)Φ_(BS)) where i=1, 2, . . . Nz, with Nz again being the total number of grids in the vertical directions, the number of vertical layers. In this extraction of Equation (17b), PI_(i) the layer productivity index is defined by Equation (16) and the Φ_(B) potential terms are the known boundary potentials at boundaries of the neighbor cells to the west, east, north and south of the central cell. As is conventional, the terms, cell and grid block are used interchangeably.

Conventional well models can be generally classified in three groups: (a) the Explicit Well Model; (b) the Bottom Hole Pressure Specified Well Model; and the Fully Implicit Well Model (Aziz K, Settari A, Petroleum Reservoir Simulation, Applied Science Publishers Ltd, London 1979). For a better understanding of the present invention, a brief review of each well model is presented.

Explicit Well Model

For an explicit well model, the source term q_(i) in Equation (10) is defined according to Equation (12) by:

$\begin{matrix} {q_{i} = {\frac{k_{x,i}\Delta\; z_{i}}{\sum\limits_{i = 1}^{i = {Nz}}{k_{x,i}\Delta\; z_{i}}}q_{T}}} & (12) \end{matrix}$ where q_(i) is the production rate for cell (grid block) i where the well is going through and perforated.

Substituting Equation (12) into Equation (10) for cell i results in

$\begin{matrix} {{{T_{Upi}\Phi_{i - 1}} + {T_{C,i}\Phi_{i}} + {T_{{Down},i}\Phi_{i + 1}}} = {b_{i}\mspace{14mu}{where}}} & (13) \\ {T_{c,i} = {{- \left( {T_{{Up},i} + T_{{down},i} + T_{Wi} + T_{Ei} + T_{Ni} + T_{Si}} \right)}\mspace{14mu}{and}}} & \left( {14a} \right) \\ {b_{i} = {{\frac{k_{x,i}\Delta\; z_{i}}{\sum\limits_{i = 1}^{i = {Nz}}{k_{x,i}\Delta\; z_{i}}}q_{T}} - \left( {{T_{Wi}\Phi_{BW}} + {T_{Ei}\Phi_{BE}} + {T_{Ni}\Phi_{BN}} + {T_{Si}\Phi_{BS}}} \right)}} & \left( {14b} \right) \end{matrix}$

Writing Equation (13) for all the cells i=1, Nz around the well for the well cells only results in a linear system of equations with a tridiagonal coefficient matrix of the type illustrated in FIG. 12, which can be written in matrix vector notation as below: A _(RR){right arrow over (Φ)}_(R) ={right arrow over (b)} _(R)  (15)

In Equation (15), A_(RR) is a (Nz×Nz) tridiagonal matrix, and Φ_(R) and b_(R) are (Nz×1) vectors. Equation (15) is solved by computer processing for the reservoir unknown potentials Φ_(R) grid blocks where well is going through by a tridiagonal linear system solver such as the Thomas algorithm.

Bottomhole Pressure Specified Well Model

Φ_(w) is the uniform potential along the wellbore open to production using conventional techniques according to techniques explained in the literature, such as the textbooks by Muskat “Physical Principles of Oil Production”, McGraw-Hill Book Co. (1949) and “The Flow of Homogeneous Fluids Through Porous Media”, McGraw-Hill Book Co. (1937). For the purposes of modeling in the present context, the friction pressure drop along the well is considered negligible. Assuming that Φ_(w) is known (or specified), the oil rate from the perforation is calculated by Equation (16) as follows:

$\begin{matrix} {q_{i} = {{{PI}_{i}\mspace{14mu}\left( {\Phi_{i} - \Phi_{W}} \right)} = {\frac{2\pi\; k_{x,i}\Delta\; z_{i}}{\ln\left( {r_{o,i}/r_{w}} \right)}\mspace{14mu}\left( {\Phi_{i} - \Phi_{W}} \right)}}} & (16) \end{matrix}$ where PI_(i) is the layer productivity index, Φw is the specified bottom hole potential (datum corrected pressure), Φi is the reservoir grid block pressure where well is going through for grid block(cell) i, r_(o,i) is called Peaceman's well block radius for grid block i defined as 0.2Δx, r_(w) is the radius of the well.

The variables in Equation (16) are explained in the Nomenclature section above. Substituting Equation (16) into Equation (10) and collecting the terms for the cell i, for cell i the following result occurs: T _(Upi)Φ_(i−1) +T _(C,i)Φ_(i) +T _(Down,i)Φ_(i+1W) =b _(i)  (17) Let T _(ci)=−(T _(Up,i) +T _(down,i) +T _(Wi) +T _(Ei) +T _(Ni) ++T _(Si)+PI_(i))  (17a) b _(i)=−(PI_(i)Φ_(W) +T _(Wi)Φ_(BW) +T _(Ei)Φ_(BE) +T _(Ni)Φ_(BN) +T _(Si)Φ_(BS))  (17b)

Equation (17) when written for all the grid blocks i=1, Nz results in a matrix system illustrated in FIG. 14. The matrix of FIG. 14 for the bottom hole pressure specified well model can be seen to be similar to the matrix of FIG. 12, and in a comparable manner Equation (17) is similar to Equation (13). The bottom hole pressure specified well model can be easily solved by matrix computer processing with a tridiagonal equation solver methodology.

Fully Implicit Well Model

Total production rate q_(T) for a well according to a fully implicit well model is specified according to Equation (18).

$\begin{matrix} {{q_{T} - {\sum\limits_{i = 1}^{i = {Nz}}{{PI}_{i}\left( {\Phi_{i} - \Phi_{W}} \right)}}} = 0.0} & (18) \end{matrix}$

The individual completion rate q_(i) is calculated by Equation (16). For the implicit well model, the wellbore potential Φ_(W), is assumed constant throughout the well but it is unknown.

Substituting Equation (18) into Equation (10) and collecting the terms for the cell i for cell (i) arrives at the following expression: T _(Upi)Φ_(i−1) +T _(C,i)Φ_(i) +T _(Down,i)Φ_(i+1W)−PI_(i)Φ_(W) =b _(i)  (19) Let T _(ci)=−(T _(Up,i) +T _(down,i) +T _(Wi) +T _(Ei) +T _(Ni)+PI_(i))  (19a) b _(i)=−(T _(Wi)Φ_(BW) +T _(Ei)Φ_(BE) +T _(Ni)Φ_(BN) +T _(Si)Φ_(BS))  (19b)

Writing Equation (19) for all cells yields a linear system of equations with the form illustrated in FIG. 13 upper diagonal solid line represents T_(Up,i) as defined by Equation (11), and the lower diagonal solid line describes the elements called T_(Down,i) described in Equation (11). The central term T_(C,i) is defined by Equation (19a) and right hand side b_(i) is defined by Equation (19b).

The linear system of the matrix (Equation 19) of FIG. 13 can be represented in vector matrix notation as below:

$\begin{matrix} {{\begin{bmatrix} A_{RR} & A_{RW} \\ A_{WR} & A_{WW} \end{bmatrix}\begin{bmatrix} {\overset{->}{\Phi}}_{R} \\ \Phi_{W} \end{bmatrix}} = \begin{bmatrix} {\overset{->}{b}}_{R} \\ b_{W} \end{bmatrix}} & (20) \end{matrix}$

In Equation (20), A_(RR) is a (Nz×Nz) tridiagonal matrix, A_(RW) is a (Nz×1) vector (reservoir PI's), A_(WR) is a (1×Nz) vector (PI's) and A_(WW) is (1×1) scalar. For this example:

$A_{ww} = {\sum\limits_{i = 1}^{N_{z}}{PI}_{NZ}}$

Writing in algebraic form: A _(RR){right arrow over (Φ)}_(R) +A _(Rw)Φ_(W) ={right arrow over (b)} _(R)  (21) A _(wR){right arrow over (Φ)}_(R) +A _(ww)Φ_(W) =b _(w)  (22)

Solving for Φ_(W) from Eq 22 results in: Φ_(W) =A ⁻¹ _(ww)(b _(w) −A _(wR){right arrow over (Φ)}_(R))  (23)

Substituting into Equations (21) A _(RR){right arrow over (Φ)}_(R) +A _(Rw)(A ⁻¹ _(ww)(b _(w) −A _(wR){right arrow over (Φ)}_(R)))={right arrow over (b)} _(R)  (24)

Collecting the terms in Equation (24) produces: (A _(RR) −A _(ww) ⁻¹ A _(wR) A _(RW)){right arrow over (Φ)}_(R) ={right arrow over (b)} _(R) −A _(Rw) A _(ww) ⁻¹ b _(w)  (25)

The coefficient matrix (A_(RR)−A_(ww) ⁻¹A_(wR)A_(RW)) of Equation (25) is an (Nz×Nz) full matrix.

The resulting coefficient matrix can be defined as follows: Ã=A _(RR) −A _(ww) ⁻¹ A _(wR) A _(RW)  (26) and {tilde over (b)}={right arrow over (b)} _(R) −A _(Rw) A _(ww) ⁻¹ b _(w)

Then Equation (26) can be written as: Ã{right arrow over (Φ)} _(r) ={tilde over (b)} _(R)  (27)

The matrix of Equation (27) can be solved by a direct solver by Gaussian elimination or any other suitable conventional solver for the full matrices.

If in the implicit well model the number of layers Nz is large, and wells are fully completed in all the layers, then the solution of Equation (27) becomes expensive in computation time. For this reason and also if many wells are involved, generally Equation (27) is solved by iterative methods. An example of this is described in “A Fully-Implicit Fully-Coupled Well Model for Parallel Mega-Cell Reservoir Simulation”, SPE Technical Symposium of Saudi Arabia Section, 14-16 May 2005.

A flow chart I (FIG. 10) indicates the basic computer processing sequence of fully implicit, fully coupled well model simultaneous solution for the type of matrix illustrated in FIG. 13. During a step 100, simulation begins by reading the reservoir and production data. Reservoir data includes reservoir geometric information-its size, extent (length) in x, y, and z directions, reservoir properties such as permeability distribution, porosity distribution, thickness of layers, relative permeability data, capillary pressure data, fluid property data such as fluid density tables, formation volume factor tables, viscosity tables, location of wells, location of well perforations within the reservoir.

Production data includes oil, water, and gas production rate measured or specified for the wells defined in the previous step. Production data also includes minimum bottomhole pressure for each well.

In many instances, only oil production rates and water production rates are input if the gas production data is not available. If no water is produced in the field, only oil production rates are read as input.

During step 102 the time step is incremented by one, and an iteration counter of the number of non-linear iterations performed during the present time step is set to zero. During step 104, a Jacobian matrix of reservoir data is formed. In step 106, the resultant linear system of Equation (19) is then solved by iterative methods using sparse preconditioners (Yousef Saad, Iterative Methods for Sparse Linear Systems, Society of Industrial & Applied Mathematics (SIAM) Publication, 2003).

During step 108, a convergence step is performed to determine whether the non-linear iterations have converged. The individual residuals of the equations resulting from step 106 are checked against user-specified tolerances. If these tolerances are satisfied, the non-linear iteration loop is exited, the solution output is written to file for the current time step and processing returns to step 102 for the time step to be advanced, and processing for the incremented time step then proceeds as indicated. If the user-specified tolerances are determined not satisfied during step 108, processing according to the non-linear iteration loop returns to step 104 and continues. If the number on non-linear iterations becomes too large, a decision may be made to reduce the time step size and go back to step 102.

However, based on the strength of the preconditioning, this method can also be very expensive in computation time, since there is no exact way of representing the full matrix in Equation (19). For difficult problems, with extreme heterogeneity and small layer thicknesses, the iterative method may not converge.

Additionally, for highly heterogeneous reservoirs with some vertically non-communicating layers, the above described well models do not produce the correct physical solution. Instead, they can produce incorrect flow profiles, and in some occasions they can cause simulator convergence problems.

The Present Invention

Explicit and fully implicit models can produce totally different flow profiles in case of some vertical flow barriers. This is displayed in FIG. 5. As shown in FIG. 5, a reservoir model V is composed of an upper layer 50 of relatively low permeability, and with vertical flow, located above an isolated high permeability layer 52 with no vertical flow communication with adjacent layers. The flow barrier layer 52 is located in the reservoir V above a layer 54 of medium permeability and having vertical flow communication. As can be seen in FIG. 5, the production rates qT for the well model V are the same for both implicit and explicit well modeling methods.

However, the production rates q1, q2, and q3 of layers 50, 52 and 54 differ significantly. The production rate q2 for the layer 52 is the major contribution to the total production rate qT as shown by curve 56 for the explicit model. On the contrary, for the implicit model as shown by curve 58, production rate q2 for the layer 52 is markedly less.

In the fully implicit well model, the production rate for the layer 2 is significantly less due to the fact that this method takes into account internal reservoir heterogeneity (all the reservoir properties throughout the reservoir) and heterogeneity around the well blocks in addition to the perforation index (or layer 2 property alone as is used by the explicit method as the only data to assign the rate fraction to this perforation). For example, the fully implicit well method sees that there is no fluid supplied into layer 2 from layer 1 and layer 3 due to the impermeable barrier between layer 2 and layer 1 and 3. Therefore, once some fluid is produced from well model layer 2, the pressure of layer 2 should go down and this layer should not supply at a high rate into the well, although the layer has very high permeability.

On the other hand, the explicit method assigns the rate for layer 2 based on the permeability of this layer alone without considering the layer connection to the above and below layers. Based on this, the explicit method will assign a very high rate for this layer and will keep it for the next simulation time step. In later simulation time steps, this will cause instability in the production rates, i.e., the simulator will reduce the time step size and will take a very long time to complete the simulation. It is unsatisfactory for reservoir simulation to have models provide divergent results for the same input data based on the modeling technique selected for use.

According to the present invention, a more comprehensive well model is provided. The well model according to the present invention is called a coupled reservoir well model. The associated numerical solution is referred to fully implicit, fully coupled and simultaneous solution. A fully implicit fully coupled reservoir well model produces correct flow profile along the perforated well interval, as will be described. As shown in FIG. 4 a reservoir model O is composed of a number z of i individual layers 1 through Nz, each with a permeability k_(x,i) and a thickness Δz_(i) and a potential Φ_(i) defined as indicated in FIG. 4, and upper and lower layers 40 of relatively low permeability, and with vertical flow, located above and below, respectively, an isolated high permeability layer 42 with no vertical flow communication with adjacent layers. As can be seen in FIG. 4, the production rates q_(i) for layer i of the model O is an indicated by the expression set forth in FIG. 4.

The fully implicit coupled reservoir well model V of FIG. 5 is presented in Equation 19 above and is also presented here in matrix format for further description:

$\begin{matrix} {{\begin{bmatrix} A_{RR} & A_{RW} \\ A_{WR} & A_{WW} \end{bmatrix}\begin{bmatrix} {\overset{->}{X}}_{R} \\ {\overset{->}{X}}_{W} \end{bmatrix}} = \begin{bmatrix} {\overset{->}{b}}_{R} \\ {\overset{->}{b}}_{w} \end{bmatrix}} & (20) \end{matrix}$

The present invention is based on the fact that the bottom hole pressure of a layered reservoir with a vertical well is the same as a system according to the present invention. The system according to the present invention is formed by identifying flow barriers and lumping together or combining for processing purposes the reservoir layers around the well which are communicating in the vertical direction. Care should be taken according to the present invention in the formation of the reduced system. The reduced system must be formed correctly, or otherwise errors in the reduced system construction can increase the total number of nonlinear Newton iterations.

The system according to the present invention is solved for the bottom hole pressure. The solution is carried out by treating the well as specified bottom hole pressure well. The procedure is fully implicit; however, it is not a simultaneous solution. Instead, the solution is sequential. The method of the present invention is convergent since it is part of the global Newton iteration of the simulator. Therefore, if the model according to the present invention is constructed correctly, any possible error in rate calculations will be small and will diminish with the simulator Newtonian iterations.

A flow chart F (FIG. 11) illustrates the basic computer processing sequence according to the present invention and the computational methodology taking place during a typical embodiment of a sequential fully implicit well model for reservoir simulation with the present invention.

During a step 200, simulation begins by reading the reservoir and production data. Reservoir and production data read in during step 200 are of the types discussed above. The reservoir simulator is also initialized during step 200, setting the simulation day and the time step to zero. During step 202 the time step is incremented by one, and an iteration counter of the number of non-linear iterations performed during the present time step is set to zero.

During step 204, a Jacobian matrix of reservoir data is formed. In step 206, a reduced system like that defined by the model R described above is formed according to the present invention and the matrix solved for bottom hole potential Φ_(w) in the manner described above. During step 208, the modified linear system matrix A is solved according to the present invention in the manner set forth above.

FIG. 15 illustrates the methodology of forming the reduced well model system matrix R and solving for bottom hole potential Φ_(w) according to steps 204 and 206 of FIG. 11. As indicated at step 210, vertical flow barriers in the original well model system are identified. This can be done based on well log data or by specification by a reservoir analyst from data in the original reservoir model.

After steps 210, a reduced well model system model is then formed by the data processing system D during a step 212. Those layers in the well model which are located between flow barrier layers and have vertical flow are combined together for analytical model purposes.

Next, during step 214, the resulting reduced well model system is solved by computer processing for bottom hole potential Φ_(w) and reservoir unknowns using the techniques discussed of Equations (17), (17a) and (17b). Step 216 then solves with the data processing system D the full well model system structural matrix of Equation (27) using a direct solver or other suitable technique described above. Completion rates q_(i) and total well flow rate q_(T) are then determined with the data processing system D based on the results of step 216.

Referring again to FIG. 11, during step 220, a convergence step is performed to determine whether the non-linear iterations have converged. The individual residuals of the equations resulting from step 216 are checked against user-specified tolerances. If these tolerances are satisfied, the non-linear iteration loop is exited, the solution output is written to file for the current time step and processing returns to step 202 for the time step to be advanced, and processing for the incremented time step the proceeds as indicated. If the user-specified tolerances are determined not satisfied during step 208, processing according to the non-linear iteration loop returns to step 204 and continues. If the number on non-linear iterations becomes too large, a decision may be made to adjust the model.

As illustrated in FIG. 16, a data processing system D according to the present invention includes a computer 240 having a processor 242 and memory 244 coupled to the processor 242 to store operating instructions, control information and database records therein. The computer 240 may, if desired, be a portable digital processor, such as a personal computer in the form of a laptop computer, notebook computer or other suitable programmed or programmable digital data processing apparatus, such as a desktop computer. It should also be understood that the computer 240 may be a multi-core processor with nodes such as those from Intel Corporation or Advanced Micro Devices (AMD), or a mainframe computer of any conventional type of suitable processing capacity such as those available from International Business Machines (IBM) of Armonk, N.Y. or other source.

The computer 240 has a user interface 246 and an output display 248 for displaying output data or records of processing of well logging data measurements performed according to the present invention to obtain a measure of transmissibility of fluid in subsurface formations. The output display 48 includes components such as a printer and an output display screen capable of providing printed output information or visible displays in the form of graphs, data sheets, graphical images, data plots and the like as output records or images.

The user interface 246 of computer 240 also includes a suitable user input device or input/output control unit 250 to provide a user access to control or access information and database records and operate the computer 240. Data processing system D further includes a database 252 stored in computer memory, which may be internal memory 244, or an external, networked, or non-networked memory as indicated at 254 in an associated database server 256.

The data processing system D includes program code 260 stored in non-transitory memories 244 of the computer 240. The program code 260, according to the present invention is in the form of computer operable instructions causing the data processor 242 to form a sequential fully implicit well model for reservoir simulation according to the present invention in the manner that has been set forth.

It should be noted that program code 260 may be in the form of microcode, programs, routines, or symbolic computer operable languages that provide a specific set of ordered operations that control the functioning of the data processing system D and direct its operation. The instructions of program code 260 may be stored in memory 244 of the computer 240, or on computer diskette, magnetic tape, conventional hard disk drive, electronic read-only memory, optical storage device, or other appropriate data storage device having a computer usable non-transitory medium stored thereon. Program code 260 may also be contained on a data storage device such as server 64 as a non-transitory computer readable medium, as shown.

Two illustrative example model problems are presented below: a seven layer homogeneous reservoir with one fracture flow barrier (FIG. 7); and a twenty-two layer heterogeneous reservoir with two fracture flow barriers (FIG. 9).

Seven Layer Homogeneous Well Model

FIG. 7A illustrates the seven reservoir layers and properties for the original model 70 and reduced model 71. As seen, it was assumed that reservoir has seven layers. Layer thickness of each layer 72 with vertical flow is 10 ft. A layer 73 which represents a fracture with no vertical flow is present which is 1 ft. thick. It was further assumed that the layer 73 does not communicate with the layers 72 above and below. It was assumed that there is a vertical well in the middle, as indicated by an arrow. Initial reservoir potential (datum corrected pressure) is 3,000 psi for the model of FIG. 7A. Each layer 72 was assumed to have 10 md areal permeability k_(x) and k_(y) and 1 md vertical permeability k_(z).

Table 1 summarizes the reservoir and grid properties for the model 70. The grid size in the area direction (square grid) was assumed to be 840 ft. Oil viscosity was set to 1 cp and Oil Formation Volume factor was assumed to be 1. Well Total Oil Production rate was set to 1,000 B/D. A Layer Productivity Index PI for each layer completion was calculated by Peaceman's method, as described, and is also shown in Table 1.

TABLE 1 Problem 1 - Reservoir Properties Layer Thickness, ft K_(x) = K_(y), md K_(z), md Layer, PI b/d/psi 1 10. 10. 1 0.12 2 10. 10. 1 0.12 3 1 10,000 1.e−9 12.14 4 10 10 1 0.12 5 10 10 1 0.12 6 10 10 1 0.12 7 10 10 1 0.12

Fully Implicit Fully Coupled Simultaneous Solution

The coefficient matrix for the solution of reservoir pressures and the bottom hole pressure is fainted in a similar manner explained above with regard to Equations (18-19) and as shown in FIG. 13. It can be seen that there are only 8 unknowns (7 potentials or datum corrected pressures) and the one bottom hole potential), and that the coefficient matrix is non-sparse. The linear system of equations can be solved by a direct method, such as Gaussian Elimination, for the unknown reservoir (layer) potentials Φ_(i), i=1,7 and the other unknown Φ_(W).

Results

Table 2 summarizes the calculated layer potentials, wellbore potential and layer (completion) flow rates for the model 70 of FIG. 7A.

TABLE 2 Exact Solution of the Original Problem Bottom hole (Wellbore) Layer Potential, psi Potential, psi Rate/d 1 2630.61 1257.36 166.67 2 2630.61 1257.36 166.67 3 1257.37 1257.36 0.0 4 2630.61 1257.36 166.67 5 2630.61 1257.36 166.67 6 2630.61 1257.36 166.67 7 2630.61 1257.36

From the computed results we see that computed bottom hole potential Φ_(W)=1257.36 psi.

Formation of the Problem According to the Present Invention

According to the reservoir data in Table 1 and as shown in FIG. 7A, there is only one layer 73 which does not communicate vertically with the other layers. Therefore, as shown in FIG. 7B, the layers 72 above the fracture layer 73 are combined into a single layer according to form the reduced well model in accordance with the present invention. Similarly layers 72 below layer 73 are combined into a single layer. The reduced model now can be seen to have only three layers. The total number of the unknowns is 4 as opposed to 8 as in the full model.

Table 3 summarizes the properties of reduced well model 71 formed according to processing with the present invention.

TABLE 3 Reduced Well Model Layer Thickness, ft K_(x) = K_(y), md K_(z), md PI, b/d/psi 1 20 10 1 0.24 2 1 10,000 0 12.14 3 40 10 1 0.40

The linear system of equations (Equation 20) for the reduced system still has an unstructured coefficient matrix, but with a 50% less number of unknowns. In actual reservoirs, with hundreds of layers and only a few flow barriers, the well model size reduction according to the present invention would be drastic, for example, a reduced well model system model according to the present invention could be 1 percent of size of the full system. The reduced system is solved by a direct solver for the layer potentials and the bottom hole potential. Table 4 presents the results.

TABLE 4 Results of the Reduced System Bottom hole (Wellbore) Layer Potential, psi Potential, psi 1 2630.61 1257.36 2 1257.37 1257.36 3 2630.61 1257.36

It can be seen that computed bottom hole potential: Φ_(W)=1257.36 psi is exactly the same as the Φ_(W) calculated for the full model.

The determined well potential is the only information needed for the next step. The well is next treated as a specified bottom hole pressure (potential) model. Computer processing according to the procedure described with the matrix of FIG. 14 and Equations (16 and 18) are followed to calculate the flow profile (layer rates) and the total well rate. In FIG. 14, upper diagonal solid line of the matrix represents T_(Up,i) as defined by Equation (11), and the lower diagonal solid line of the matrix describes the elements called T_(Down,i) as also defined by Equation (2). The central term T_(C,i) is defined by Equation (17a) and the right hand side hi defined by Equation (17b).

The results are summarized in Table 5. It is to be noted that total calculated well rate is exactly the same as input value of 1,000b/d.

TABLE 5 Results for the Total System with the Present Invention Layer Potential, psi Rate/d 1 2630.61 166.67 2 2630.61 166.67 3 1257.37 0.0 4 2630.61 166.67 5 2630.61 166.67 6 2630.61 166.67 7 2630.61 166.67 Total 1,000.

Results presented in Table 5 are the same as in Table 1 for the fully implicit well model. Difference or error between well rates for the calculated and input well is zero for this case and there no need for an extra iteration. This is because of the fact that the reservoir was homogeneous and no upscaling errors were made while forming the reduced system. Matrix diagonal elements and right hand side are the same as in FIG. 14, i.e., lower diagonal solid line represents T_(up,i) defined by Equation (11), upper diagonal solid line describes the elements called T_(Down,i) described above. The central term T_(C,i) defined by Equation (17a) and right hand side b, defined by Equation (17b).

The terms PI appear on Equation (16) are the perforation productivity indexes for a square grid is defined by:

${PI}_{i} = {2\pi\; k_{x,i}\frac{\Delta\; z_{i}}{\ln\left( {0.2\Delta\;{x/r_{w}}} \right)}}$ where r_(w) is the well bore radius.

Comparison with the Explicit Well Model

In several reservoir simulators, semi implicit well models or explicit well models are used. If the formulation of the well model is semi-implicit but it collapses to explicit in the pressure variable, this formulation collapses to explicit well models. The explicit well model is for this problem obtained by following the computer processing procedures for the matrix of FIG. 12 and Equations (12-14). In FIG. 12, the terms T_(Down,i), T_(Up,i) which appear on the diagonal elements are defined by Equation (11), and T_(c,i), b_(i) are defined by Equation (14a) and Equation (14b). FIG. 8A illustrates the seven reservoir layers and properties for an implicit well model 80 and FIG. 8B an explicit well model 81, of like structure to the model of FIGS. 7A and 7B. As seen it was assumed that reservoir has seven layers. Layers 82 each have a potential Φ of 2630 psi. A layer 83 which represents a fracture and was further assumed that does not communicate with the layers 72 above and below has a potential Φ of 1257 psi. It was assumed that there is a vertical well in the middle, as indicated by an arrow. FIGS. 8A and 8B compare the results of Implicit and Explicit Models. Computed perforation (layer) rates are summarized in Table 6.

TABLE 6 Comparison of Perforation (Layer) Rates for Different Well Models Exact Solution New Explicit Layer/ (Fully Implicit Fully Coupled Method Method Perforation Simultaneous Solution), Rate b/d Rate, b/d Rate, b/d 1 166.67 166.67 9.43 2 166.67 166.67 9.73 3 0.0 0.0 943.40 4 166.67 166.67 9.43 5 166.67 166.67 9.43 6 166.67 166.67 9.43 7 166.67 166.67 9.43

As can be seen, the model 81 according to the explicit method is inaccurate; it totally miscalculates the perforation rate. The explicit model method assigns practically all the well production from the thin fracture layer 83 as indicated in FIG. 8B since this layer has the highest productivity index.

The implicit methods (whether the computationally intensive fully implicit model or the reduced model according to the present invention) do not make such an assignment and instead determine that the layer 83 is not getting fluid support from the layers 82 above and below. The only fluid support a fracture layer of this type shown at 83 when present in an actual reservoir can get is from its planar neighboring cells. However, since the fracture layer is a very thin layer, transmissibility in these directions is by nature small. Therefore, the fracture layer cannot supply fluid at the rates simulated by the explicit model.

In fact, conventional implicit well models show that during the transient time the fracture layers support most of the well production as do the explicit methods. However, the layer pressure in layer 83 quickly declines and assumes the value of the uniform wellbore potential (constant bottom hole pressure). After the pressure declines, it reaches steady state, and the well production rate is in fact made by the contribution from the layers 82 above and below the vertical flow barrier 83.

Twenty-Two Layer Heterogeneous Reservoir Model

A model grid system 90 includes twenty two layers as shown in FIG. 9. The location of high permeability fracture layers 6 and 12 as counted moving downward through the layer and indicated schematically at 91 and 92. There are five layers 93, numbered 1 through 5 above layer 91, each with vertical flow. There are also five layers 94 in the model 90 with vertical flow between the flow barrier layers 91 and 92, and ten layers 95 with vertical flow located below the flow barrier layer 92. Reservoir data for the model 90 is shown in Table 7.

TABLE 7 Reservoir Data for 22 Layer Problem Thickness, Permeability, mD Vertical Permeability, Layer ft (K_(x) = K_(y)) K_(z), mD 1 10 2 2 2 10 5 1 3 10 3 3 4 10 10 5 5 10 5 4 6 1 1,000. 1.e−9 7 10 6 6 8 10 3 3 9 10 9 6 10 10 12 2 11 10 5 5 12 1 1,000. 1.e−9 13 10 7.5 3.5 14 10 7.5 3.5 15 10 7.5 3.5 16 10 7.5 3.5 17 10 7.5 3.5 18 10 9.2 1.2 19 10 9.2 1.2 20 10 9.2 1.2 21 10 9.2 1.2 22 10 9.2 1.2

-   -   Areal neighboring cell permeabilities=20 mD     -   Total Well Production Rate=2,500 B/D     -   Well Completed in all layers.     -   Results:     -   Fully Implicit Fully Coupled Simultaneous Solution     -   Calculated bottom hole potential Φ_(W)=1421.247 psi

TABLE 8 Potential Distribution, psi Layer Φ_(W) Φ_(i) 1 1421.25 2901.32 2 1421.25 2900.82 3 1421.25 2900.38 4 1421.25 2900.38 5 1421.25 2900.38 6 1421.25 1421.25 7 1421.25 2864.43 8 1421.25 2864.37 9 1421.25 2864.10 10 1421.25 2863.88 11 1421.25 2864.03 12 1421.25 1421.25 13 1421.25 2841.62 14 1421.25 2841.58 15 1421.25 2841.48 16 1421.25 2841.33 17 1421.25 2841.13 18 1421.25 2840.65 19 1421.25 2840.08 20 1421.25 2839.65 21 1421.25 2839.37 22 1421.25 2839.24

The model 90 was then subjected to the explicit model techniques of the type described above and flow data determined. A comparison of flow rate distribution for fully implicit and explicit processing of the reservoir model 90 using techniques previously described is set forth in Table 9.

TABLE 9 Comparison of Flow Rates for Fully Implicit Fully Coupled and Explicit Well Methods Layer Implicit Explicit 1 35.93 14.56 2 89.78 36.39 3 53.85 21.83 4 179.48 72.78 5 89.74 36.39 6 0.00 727.80 7 105.09 43.67 8 52.54 21.83 9 157.60 65.50 10 210.10 87.34 11 87.55 36.39 12 0.00 727.80 13 129.29 54.59 14 129.28 54.59 15 129.28 54.59 16 129.26 54.59 17 129.24 54.59 18 158.49 66.96 19 158.42 66.96 20 158.37 66.96 21 158.34 66.96 22 158.33 66.96

Reduced Model Construction

Since there are only two vertical flow barriers layers 91 and 92, layers 93 above layer 91 in FIG. 9 can be combined into one layer; layers 94 below layer 91 into one layer, and layers 95 below layer 92 into another single layer. Therefore the total number of layers according to the present invention is 5. The properties of the reduced model are as follows:

TABLE 10 Reduced Well Model Properties Layer Thickness, ft K_(x), mD K_(z), mD PI, b/d/psi PI Fraction 1 50 5 2.19 0.3 0.07 2 1 1000 0 1.21 0.29 3 50 7 3.66 0.42 0.10 4 1 1000 0 1.21 0.29 5 100 8.35 1.79 1.01 0.24

Calculated Bottom Hole Potential

Φ_(W)=1421.34 psi

The reduced model in accordance with the present invention the proceeds to determine bottom hole potential Φ_(w). The results of the reduced model with five layers are set forth below in Table 11.

TABLE 11 Potential Distribution Layer Pot wf Pot 1 1421.34 2900.53 2 1421.34 1421.35 3 1421.34 2864.16 4 1421.34 1421.35 5 1421.34 2740.61

Using the bottom hole Potential Φ_(w) calculated from the reduced model and computing potentials using specified bottom hole pressure Φ_(W) for the full model, completion layer rates are calculated according to Equation (16). The results are indicated below in Table 12.

TABLE 12 Calculated Well Layer Rates Layer New Method Fully Coupled Method 1 35.92 35.93 2 89.78 89.78 3 53.85 53.48 4 179.47 179.47 5 89.73 89.73 6 0.00 0.00 7 105.09 105.09 8 52.54 52.54 9 157.59 157.60 10 210.09 210.10 11 87.55 87.55 12 0.00 0.00 13 129.28 129.29 14 129.28 129.28 15 129.27 129.28 16 129.25 129.26 17 129.24 129.24 18 158.48 158.49 19 158.41 158.42 20 158.37 158.37 21 158.33 158.34 22 158.32 158.33

Newly  calculated  q_(t) = 2499.84  b/d Error = 2, 500. − 2499.8488 = 0.15  b/d

Error in the total rate and computed bottom hole pressure vanishes with the simulator's non-linear Newton iterations. The present invention obtains a reduced model with a production rate acceptably accurate in comparison to the results obtained by the fully implicit, fully coupled processing techniques of the prior art, but with a substantial reduction in model complexity and computer processing time.

The present invention, as has been described above, does not require a special linear solver for the solution of the coupled reservoir and well equations. In contrast, the coefficient matrix for the previously used coupled reservoir and well equations does not have regular sparse structure. Therefore, the conventional types of coupled reservoir and well equations require special solvers which can be expensive and can also face convergence problems.

It can be seen that, as described above, the present invention does not require any special solver for the solution of coupled reservoir and well equations. The same solver used for reservoir equations is utilized. The only modification made to the coefficient matrix is in the diagonal terms.

The present invention solves reservoir simulation problems where the vertical wells have many completions, which is a common occurrence in reservoirs. In recent simulation studies wells with more than 100 vertical layers (completions) are very common. The fully coupled fully implicit well model with simultaneous solution is very expensive for these cases. The present invention can save significant amounts of computer time.

The present invention is very useful for wells having hundreds of perforations completed in highly heterogeneous reservoirs. The present invention reduces the large, time consuming problem of well modeling simulation in reservoirs with large numbers of layers (completions) problem to a small problem by recognizing and advantageously using the physical principles involved. With the present invention, it has been found that vertically communicating layers can be lumped into a single layer. The reduced model so formed preserves the same bottom hole pressure as the original full model. Once the reduced model is solved for the bottom hole pressure, the wells in the large system are then treated as specified bottom hole pressure and solved easily by a conventional linear solver. Thus, the present invention eliminates the need for writing or acquiring unstructured linear solvers for many wells with hundreds of completions, which would be expensive.

The invention has been sufficiently described so that a person with average knowledge in the matter may reproduce and obtain the results mentioned in the invention herein Nonetheless, any skilled person in the field of technique, subject of the invention herein, may carry out modifications not described in the request herein, to apply these modifications to a determined structure, or in the manufacturing process of the same, requires the claimed matter in the following claims; such structures shall be covered within the scope of the invention.

It should be noted and understood that there can be improvements and modifications made of the present invention described in detail above without departing from the spirit or scope of the invention as set forth in the accompanying claims. 

The invention claimed is:
 1. A computer implemented method of forming a model of determined well production rate of component fluids from a well in a subsurface reservoir and of determined layer completion rates of perforated well intervals in the well, from measured total well production, with a coupled well reservoir model during reservoir simulation of well production at a time step during a plurality of time steps of the reservoir simulation, the coupled well reservoir model being organized into a reservoir grid subdivided into: (1) a plurality of finite difference reservoir cells for the reservoir simulation comprising a plurality of reservoir neighbor cells at the perforated well intervals; (2) the perforated well intervals in the reservoir being located at a plurality of formation layers; (3) the perforated well intervals of the formation layers having unknown well potentials, transmissibilities and fluid completion rates for component fluids at the time step; (4) the formation layers comprising vertical fluid flow layers having vertical fluid flow therefrom and flow barrier layers with no vertical fluid flow therefrom; (5) the formation layers each further having a permeability, a transmissibility, a thickness and a layer potential; (6) the reservoir grid further having a plurality of well cells at locations of the well in the grid adjacent reservoir neighbor cell of the well in the formation layers of the reservoir grid, the computer implemented method determining layer completion rates for the component fluids from the formation layers of the well and well production rates of the component fluids from the well, the computer implemented method comprising the steps of: (a) forming a full computation matrix reservoir model of reservoir data of cells of the reservoir grid, including the reservoir data for the reservoir neighbor cells at the perforated well intervals of the plurality of formative layers, the reservoir data including the permeability, thickness, and potential of each of the plurality of formation layers; (b) forming a reduced well model system matrix of reservoir data by assembling as single vertical flow layers in the matrix the data of the vertical fluid flow layers which have vertical fluid flow therebetween and are located between flow barrier layers in the reservoir model; (c) determining a bottomhole pressure for the well by solving the reduced well model system matrix for the bottom hole pressure of the well; (d) forming a coupled reservoir well model, treating the well as a bottomhole, pressure specified well having the determined bottomhole pressure as a constant pressure along the well, wherein the coupled reservoir well model is in the form of a matrix: [T _(ci)][Φ_(i) ]=[{tilde over (b)} _(i)] wherein T_(ci) is a matrix of the transmissibilities of the reservoir neighbor cells to the well cells at the perforated well intervals; Φ_(i) is a vector of unknown reservoir potentials for the reservoir neighbor cells; and {tilde over (b)}_(i) is a vector of the well net inflow data for the well from the reservoir neighbor cells; (e) solving the coupled reservoir well model to determine the fluid flows in the reservoir neighbor cells of the formation layers and the transmissibilities and potentials of the reservoir neighbor cells at the time step for each of the formation layers; (f) solving the coupled reservoir well model to determine the productivity indexes of the reservoir neighbor cells and well cells at the perforated well intervals of the reservoir at the time step; (g) determining layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well based on the determined productivity indexes of the reservoir neighbor cells and well cells at the perforated well intervals of the reservoir at the time step; (h) determining total well production rate for the well from the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well at the time step; and (i) forming a record of the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well and the determined total well production rate for the well at the time step.
 2. The computer implemented method of claim 1, wherein the matrix of the transmissibilities of the reservoir neighbor cells comprises a tridiagonal matrix representing the transmissivities of the cells of the reservoir.
 3. The computer implemented method of claim 1, wherein the steps of solving the coupled well reservoir model comprise applying a full matrix solver.
 4. The computer implemented method of claim 3, further including the steps of: comparing residuals computed from the step of applying a full matrix solver against specified tolerances; and if the tolerances are satisfied at the time step, and for each time step during life of the subsurface reservoir, proceeding to the step of forming a record; and if the tolerances are not satisfied at the time step, returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the time step.
 5. The computer implemented method of claim 3, further including the steps of: comparing residuals computed from the step of applying a full matrix solver against specified tolerances; and if the specified tolerances are satisfied at the time step, but not for each time step during life of the reservoir, incrementing the simulator time step; and returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the incremented simulator time step.
 6. The computer implemented method of claim 1, wherein the step of solving the coupled well reservoir model comprises the step of solving for fluid flows based on the permeability, thickness and layer potentials of the formation layers.
 7. A data processing system forming a model of determined well production rate of component fluids from a well in a subsurface reservoir and of determined layer completion rates of perforated well intervals in the well, from measured total well production, with a coupled well reservoir model during reservoir simulation of well production at a time step during a plurality of time steps of the reservoir simulation, the coupled well reservoir model being organized into a reservoir grid subdivided into: (1) a plurality of finite difference reservoir cells for the reservoir simulation comprising a plurality of reservoir neighbor cells at the perforated well intervals; (2) the perforated well intervals in the reservoir being located at a plurality of formation layers; (3) the perforated well intervals of the formation layers having unknown well potentials, transmissibilities and fluid completion rates for component fluids at the time step; (4) the formation layers comprising vertical fluid flow layers having vertical fluid flow therefrom and flow barrier layers with no vertical fluid flow therefrom; (5) the formation layers each further having a permeability, a transmissibility, a thickness and a layer potential; and (6) the reservoir grid further having a plurality of well cells at locations of the well in the formation layers of the reservoir grid, the data processing system determining layer completion rates for the component fluids from the formation layers of the well and well production rates of the component fluids from the well, the data processing system comprising: a processor performing the steps of: (a) forming a full computation matrix reservoir model of reservoir data of cells of the reservoir grid, including the reservoir data for the reservoir neighbor cells at the perforated well intervals of the plurality of formation layers, the reservoir data including the permeability, thickness and potential of each of the plurality of formation layers; (b) forming a reduced well model system model matrix of reservoir data by assembling as single vertical flow layers in the matrix the data of the vertical fluid flow layers which have vertical fluid flow therebetween and are located between flow barrier layers in the reservoir model; (c) determining a bottomhole pressure for the well by solving the reduced well model system matrix for the bottom hole pressure of the well; (d) forming a coupled reservoir well model, treating the well as a bottomhole, pressure specified well having the determined bottomhole pressure as a constant pressure along the well, wherein the coupled reservoir well model is in the form of a matrix: [T _(ci)][Φ_(i) ]=[{tilde over (b)} _(i)] wherein T_(ci) is a matrix of the transmissibilities of the reservoir neighbor cells to the well cells at the perforated well intervals; Φ_(i) is a vector of unknown reservoir potentials for the reservoir neighbor cells; and {tilde over (b)}_(i) is a vector of the well net inflow data for the well from the reservoir neighbor cells; (e) solving the coupled reservoir well model to determine the fluid flows in the reservoir neighbor cells of the formation layers and the transmissibilities and potentials of the reservoir neighbor cells at the time step for each of the formation layers; (f) solving the coupled reservoir well model to determine the productivity indexes of the reservoir neighbor cells and well cells at the perforated well intervals of the reservoir at the time step; (g) determining layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well based on the determined productivity indexes of the reservoir neighbor cells and well cells at the perforated well intervals of the reservoir at the time step; (h) determining total well production rate for the well from the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well at the time step; and a memory performing the step of forming a record of the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well and the determined total well production rate for the well at the time step.
 8. The data processing system of claim 7, wherein matrix of the transmissibilities of the reservoir neighbor cells comprises a tridiagonal matrix representing the transmissivities of the cells of the reservoir.
 9. The data processing system of claim 7, wherein the processor in solving the coupled well reservoir model performs the step of applying a full matrix solver.
 10. The data processing system of claim 7, further including the processor performing the steps of: comparing residuals computed from the step of applying a full matrix solver against specified tolerances; and if the tolerances are satisfied at the time step, and for each time step during life of the subsurface reservoir, causing the memory to perform the step of forming a record; and if the tolerances are not satisfied at the time step, returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the time step.
 11. The data processing system of claim 7, further including the processor performing the steps of: comparing residuals computed from the step of applying a full matrix solver against specified tolerances; and if the specified tolerances are satisfied at the time step, but not for each time step during life of the reservoir, incrementing the simulator time step; and returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the incremented simulator time step.
 12. The data processing system of claim 7, wherein the processor in solving the coupled well reservoir model performs the step of solving for fluid flows based on the permeability, thickness and layer potentials of the formation layers.
 13. A data storage device having stored in a non-transitory computer readable medium computer operable instructions for causing a data processor, in forming a model of determined well production rate of component fluids from a well in a subsurface reservoir and of determined layer completion rates of perforated well intervals in the well, from measured total well production, with a coupled well reservoir model during reservoir simulation of well production at a time step during a plurality of time steps of the reservoir simulation, the coupled well reservoir model being organized into a reservoir grid subdivided into: (1) a plurality of finite difference reservoir cells for the reservoir simulation comprising a plurality of reservoir neighbor cells at the perforated well intervals; (2) the perforated well intervals in the reservoir being located at a plurality of formation layers; (3) the perforated well intervals of the formation layers having unknown well potentials, transmissibilities and fluid completion rates for component fluids at the time step; (4) the formation layers comprising vertical fluid flow layers having vertical fluid flow therefrom and flow barrier layers with no vertical fluid flow therefrom; (5) the formation layers each further having a permeability, a transmissibility, a thickness and a layer potential; (6) the reservoir grid further having a plurality of well cells at locations of the well in the grid adjacent reservoir neighbor's cells of the well in the formation layers of the reservoir grid, the stored computer operable instructions causing the data processor to determine layer completion rates for the component fluids from the formation layers of the well and well production rates of the component fluids from the well, by performing the steps of: (a) forming a full computation matrix reservoir model of reservoir data of cells of the reservoir grid, including the reservoir data for the reservoir neighbor cells at the perforated well intervals of the plurality of formation layers, the reservoir data including the permeability, thickness and potential of each of the plurality of formation layers; (b) forming a reduced well model system matrix of reservoir data by assembling as single vertical flow layers in the matrix the data of the vertical fluid flow layers which have vertical fluid flow therebetween and are located between flow barrier layers in the reservoir model; (c) determining a bottomhole pressure and the formation pressures and saturations for the well by solving the reduced well model system matrix for the bottom hole pressure of the well; (d) forming a coupled reservoir well model, treating the well as a bottomhole, pressure specified well having the determined bottomhole pressure as a constant pressure along the well, wherein the coupled reservoir well model is in the form of a matrix: [T _(ci)][Φ_(i) ]=[{tilde over (b)} _(i)] wherein T_(ci) is a matrix of the transmissibilities of the reservoir neighbor cells to the well cells at the perforated well intervals; Φ_(i) is a vector of unknown reservoir potentials for the reservoir neighbor cells; and {tilde over (b)}_(i) is a vector of the well net inflow data for the well from the reservoir neighbor cells; (e) solving the coupled reservoir well model to determine the fluid flows in the reservoir neighbor cells of the formation layers and the transmissibilities and potentials of the reservoir neighbor cells at the time step for each of the formation layers; (f) solving the coupled reservoir well model to determine the productivity indexes of the reservoir neighbor cells and well cells at the perforated well intervals of the reservoir at the time step; (g) determining layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well based on the determined productivity indexes of the reservoir neighbor cells and well cells at the perforated well intervals of the reservoir at the time step; (h) determining total well production rate for the wells from the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well at the time step; and (i) forming a record of the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the wells and the determined total well production rate for the well at the time step.
 14. The data storage device of claim 13, wherein the instructions for causing the processor to perform the steps of solving the coupled well reservoir model comprise instructions for applying a full matrix solver.
 15. The data storage device of claim 14, wherein the instructions further include instructions causing the processor to perform the steps of: comparing residuals computed from the step of applying a full matrix solver against specified tolerances; and if the specified tolerances are satisfied at the time step, but not for each time step during life of the reservoir, incrementing the simulator time step; and returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the incremented simulator time step.
 16. The data storage device of claim 13, wherein the instructions further include instructions causing the processor to perform the steps of: comparing residuals computed from the step of applying a full matrix solver against specified tolerances; if the tolerances are satisfied at the time step, and for each time step during life of the reservoir, proceeding to the step of forming a record; and if the tolerances are not satisfied at the time step, returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the time step.
 17. The data storage device of claim 13, wherein the instructions for causing the processor to perform the step of solving the coupled well reservoir model comprise instructions for solving for fluid flows based on the permeability, thickness and layer potentials of the formation layers. 